Reasoning · Grade 4-2 Adding and Subtracting Fractions

Problem

Find the numbers the symbols stand for

Two mixed numbers in sixths add up to 8 and 2 sixths. The first has whole part (dot) and fraction 3 sixths. The second has whole part (triangle) and fraction (square) sixths, and the three symbols differ. Find the largest and smallest three-digit number they can spell.
Your answer
How to solve
Strategy Make a Systematic List — A mixed-number sum splits cleanly into two smaller questions: what happens with the sixths, and what happens with the whole numbers. Working backwards from the answer 8 and 2/6, the sixths alone pin down (square) with no guessing at all, and they also tell me whether a whole was carried. Once the carry is known, the whole-number part becomes a single easy statement about (dot) + (triangle), and I can make a short, complete list of the pairs that fit. With a list that is guaranteed complete, picking the greatest and the least three-digit number is just reading the list.
1STEP 1

Split the sum into sixths and whole numbers

Split the sum into whole parts and sixths.

●3/6 + ▲■/6 = (● + ▲) + (3 + ■)/6
2STEP 2

Work backwards from the 2 sixths to find (square)

To leave 2 sixths after 3 sixths, the square is 5.

3/6 + 5/6 = 8/6 = 1 2/6
3STEP 3

Use the carry to fix the whole-number total

One whole carries over, so the two whole parts add to 7.

● + ▲ + 1 = 8 → ● + ▲ = 7
4STEP 4

List every pair of different numbers adding to 7

Excluding 5, the pairs are 1 and 6, 3 and 4, 4 and 3, 6 and 1.

7 = 1+6 = 2+5 = 3+4 = 4+3 = 5+2 = 6+1
5STEP 5

Read off the three-digit numbers and compare them

Read as numbers, 615 is largest and 165 smallest.

165 ＜ 345 ＜ 435 ＜ 615
Answer
615, 165
dot + triangle = 7, square = 5
Both winners can be plugged straight back in. For 615: 6 and 3/6 plus 1 and 5/6 gives 7 wholes and 8 sixths, and 8 sixths is 1 whole and 2 sixths, so the total is 8 and 2/6. For 165: 1 and 3/6 plus 6 and 5/6 gives the same 7 wholes and 8 sixths, again 8 and 2/6. The sizes are believable too. Every answer must end in 5, so the greatest one needs the biggest possible hundreds digit; (dot) cannot be 7 or more, because then (triangle) would have to be 0 or negative, so 6 is the ceiling and 615 is right. Likewise the smallest hundreds digit allowed is 1, giving 165.
Takeaway

Add the fraction parts first: they tell you the missing numerator and whether a whole gets carried, and everything else falls out of that.

  • Split the sum into sixths and whole numbers
  • Work backwards from the 2 sixths to find (square)
  • Use the carry to fix the whole-number total
  • List every pair of different numbers adding to 7
  • Read off the three-digit numbers and compare them